CONVERGENCE OF COMPLEX MULTIPLICATIVE CASCADES

• 成果类型: Article
• 署名作者: Barral, Julien; Jin, Xiong; Mandelbrot, Benoit
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris 13; Inria; Yale University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/09-AAP665
• 发表日期: 2010
• 页码: 1219-1252
• 关键词: uniform-convergence; multifractal products; LIMIT-THEOREMS; martingales; continuity; dimension; exponents; spectrum; moments; Markov

摘要

The familiar cascade measures are sequences of random positive measures obtained on [0, 1] via b-adic independent cascades. To generalize them, this paper allows the random weights invoked in the cascades to take real or complex values. This yields sequences of random functions whose possible strong or weak limits are natural candidates for modeling multifractal phenomena. Their asymptotic behavior is investigated, yielding a sufficient condition for almost sure uniform convergence to nontrivial statistically self-similar limits. Is the limit function a monofractal function in multifractal time? General sufficient conditions are given under which such is the case, as well as examples for which no natural time change can be used. In most cases when the sufficient condition for convergence does not hold, we show that either the limit is 0 or the sequence diverges almost surely. In the later case, a functional central limit theorem holds, under some conditions. It provides a natural normalization making the sequence converge in law to a standard Brownian motion in multifractal time.